Talk:Covariant derivative

curvature, parallel transport, geodesics

I removed the last two subsections (curvature and parallel transport-geodesic) they do not add anything to the correspondent articles and badly written.

The subsection on Levi-Civita connection is moved to Fundamental theorem of Riemannian geometry.

(I just realized that part of it can be used in lie bracket, but will do it next time) Tosha 20:47, 20 Jul 2004 (UTC)

∇ vs D

I think to change from D to ∇, it will make it consistent with other articles. Tosha 13:24, 21 Jul 2004 (UTC)

I agree that we should stick to one standard. I'm going to edit the page for spelling/grammar in a moment... - Gauge 02:50, 3 Aug 2004 (UTC)

not all vectors are covariant

Wiki nowadays says "A vector may be described as a list of numbers in terms of a basis, but as a geometrical object a vector retains its own identity regardless of how one chooses to describe it in a basis. This persistence of identity is reflected in the fact that when a vector is written in one basis, and then the basis is changed, the components of the vector transform according to a change of basis formula. Such a transformation law is known as a covariant transformation."

This is not correct, not all vectors are covariant. In fact, position vector, the most basic one, is usually contravariant.

A vector (i.e. a first order tensor) is neither covariant nor contravariant, it is invariant; that is the whole point of tensors. As you say the vector is independent of the basis you describe it in. You can describe the vector with contravariant coordinates in a covariant basis, or with covariant coordinates in a contravariant basis. The reason that basis vectors "change" under a coordinate transformation is that you switched to a different set of basis vectors; the original basis vectors themselves didn't change. Also, calling a set of coordinates by themselves a vector (what you probably mean by a "position vector"), while quite common especially in physics, in my opinion just confuses everyone (in the context of differential geometry at least; in euclidean geometry you can mostly get away with it). -P

92.34.140.23 (talk) 10:14, 9 June 2022 (UTC) P

Don't basis vectors always transform covariantly by definition? And a vector can then either transform with the same transformation map or its inverse i.e. either covariantly or contravariantly? Codeinpappi (talk) 12:56, 1 May 2023 (UTC)

A little more editing needed here

The section Informal definition using an embedding into Euclidean space ends its last mathematical statement as follows:

"... and yields the Christoffel symbols for the Levi-Civita connection in terms of the metric:"

$$g_{kl}{\Gamma ^{k}}_{ij}={\frac {1}{2}}\left({\frac {\partial g_{jl}}{\partial x^{i}}}+{\frac {\partial g_{li}}{\partial x^{j}}}-{\frac {\partial g_{ij}}{\partial x^{l}}}\right).$$

But whoever wrote that only provided an equation that the Christoffel symbols satisfy.

That it not the same as "yielding" the Christoffel symbols.

It seems to me that it will be helpful to use the inverse matrix (g_kl)^-1 of (g_kl) to try to isolate the Christoffel symbols.

I hope someone knowledgeable about this subject can fix this, so we really have a definition of each individual Christoffel symbol, isolated on the left side of an equation, in terms of the derivatives of the metric tensor and other things.

Fixed equation 1.

In the section "Vector fields", I have fixed equation 1. so that it is not written with vectors (or vector fields) *immediately to the left* of functions that the vector fields do not operate on.

Instead, the functions just multiply the vectors. In order to write that unambiguously, the functions need to be on the left of the vectors.

Recently added citation of writing of Niccolai, Edoardo

I recommend removal of the citation to a writing of Niccolai, Edoardo (recently added by JeppOne (talk · contribs)), unless it can be supported as reliable and noteworthy.

Please note that there are similar discussions at Talk:Bounded mean oscillation § Recently added citation of writing of Niccolai, Edoardo, Talk:Einstein–Cartan theory § Recently added citation of writing of Niccolai, Edoardo, and Talk:Covariant derivative § Recently added citation of writing of Niccolai, Edoardo. There is also discontinued discussion at User talk:JeppOne § Articles by Niccolai, Edoardo. —Quantling (talk | contribs) 13:41, 1 May 2023 (UTC)

I'm just trying to let people know that there are some important writings by my professor. That's all. And I do it for the love of science. Maybe you should inquire about the person in question before "triggering" an unfair censorial judgment, which is to the detriment of the knowledge itself.

Thank you very much JeppOne (talk) 11:57, 2 May 2023 (UTC)

Hyphen in "coordinate-free language"

I edited the article to add a hyphen because

• "Coordinate free language" (no hyphen) means a "free language" that is of type "coordinate"
• "Coordinate-free language" (with hyphen) means a "language" that is of type "coordinate free"

and we mean the latter. —Quantling (talk | contribs) 14:24, 26 July 2023 (UTC)